Question: Solve for $x$ and $y$ using elimination. $\begin{align*}5x-6y &= 4 \\ -x-3y &= 9\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $2$ $\begin{align*}-5x+6y &= -4\\ -2x-6y &= 18\end{align*}$ Add the top and bottom equations. $-7x = 14$ Divide both sides by $-7$ and reduce as necessary. $x = -2$ Substitute $-2$ for $x$ in the top equation. $5( -2)-6y = 4$ $-10-6y = 4$ $-6y = 14$ $y = -\dfrac{7}{3}$ The solution is $\enspace x = -2, \enspace y = -\dfrac{7}{3}$.